The virtual synchronous generator (VSG) is an important concept and primary control method in modern power systems. The penetration of power-electronics-based distributed generators in the power grid provides uncertainty and reduces the inertia of the system, thus increasing the risk of instability when disturbance occurs. The VSG produces virtual inertia by introducing the dynamic characteristics of the synchronous generator, which provides inertia and becomes a grid-forming control method. The disadvantages of the VSG are that there are many parameters to be adjusted and its operation process is complicated. However, with the rapid development of artificial intelligence (AI) technology, the powerful adaptive learning capability of AI algorithms provides potential solutions to this issue. Two research hotspots are deep learning (DL) and reinforcement learning (RL). This paper presents a comprehensive review of these two techniques combined with VSG control in the energy internet (EI). Firstly, the basic principle and classification of the VSG are introduced. Next, the development of DL and RL algorithms is briefly reviewed. Then, recent research on VSG control based on DL and RL algorithms are summarized. Finally, some main challenges and study trends are discussed.